We pick any region we want and integrate both sides of each equation over that region:

On the left-hand sides we can use the divergence theorem, while the right sides can simply be evaluated:

where is the total charge contained within the region . Gauss’ law tells us that the flux of the electric field out through a closed surface is (basically) equal to the charge contained inside the surface, while Gauss’ law for magnetism tells us that there is no such thing as a magnetic charge.

Faraday’s law was basically given to us in integral form, but we can get it back from the differential form:

We pick any surface and integrate the flux of both sides through it:

On the left we can use Stokes’ theorem, while on the right we can pull the derivative outside the integral:

where is the flux of the magnetic field through the surface . Faraday’s law tells us that a changing magnetic field induces a current around a circuit.

A similar analysis helps with Ampère’s law:

We pick a surface and integrate:

Then we simplify each side.

where is the flux of the electric field through the surface , and is the total current flowing through the surface . Ampère’s law tells us that a flowing current induces a magnetic field around the current, and Maxwell’s correction tells us that a changing electric field behaves just like a current made of moving charges.

We collect these together into the integral form of Maxwell’s equations:

Because Maxwell was a thug, and saw the relations underlying all of them. Gauss was extremely gifted, Ampere almost had it right, and Faraday did his own thing, but Maxwell connected it all. All the laws have their own names, individually, for the people who did formulate them originally.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.